TUTORIAL: LINE INTEGRALS AND WORK
? TUTORIAL: LINE INTEGRALS AND WORK ?
Part 1 – Taking a Line Integral
Consider the vector field equation below:
F~ = yx̂
i. Compute the line integral,
R
F~ · d~l, from h0, 1i to h1, 0i using two paths:
1. a straight line path from h0, 1i to h1, 0i and
2. a piece-wise path that travels down the +y-axis to the origin, then makes a 90 turn
and travels along the +x-axis.
How do the results compare?
Tutorial 4, Week 5
University of Colorado – Boulder
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TUTORIAL: LINE INTEGRALS AND WORK
Consider the vector field equation below:
F~ =
xx̂
y ŷ
ii. Sketch the vector field in the box below.
iii. Can you think of a physical situation that might produce this vector field?
R
iv. Choose a path to compute the line integral, F~ · d~l, from h0, 1i to h1, 0i. How did you
choose your path? Does it matter which path you choose? Discuss with your group.
PAUSE, check your results with your instructor.
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University of Colorado – Boulder
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TUTORIAL: LINE INTEGRALS AND WORK
Part 2 – Choosing an appropriate path
The line integral between to points in a vector field that has no curl (i.e., r ⇥ f~ = ~0) is
independent of the path you choose. This doesn’t mean that the line integral is zero, just
that its value only depends on the location of the end points.
i. Below are plots of several vector fields which have no curl. Two points, labeled A and B,
appear in each diagram. Sketch the best choice of path for the given field if you were asked
to compute the line integral from A to B. Do not actually compute the line integral.
f~ = x̂
A: h 2, 2i B: h1, 3i
f~ = y ŷ
A: h 2, 3i B: h2, 1i
f~ = xx̂ + y ŷ p
A: h 2, 2i B: h0, 8i
f~ = yx̂ + xŷ
A: h 3, 3i B: h3, 3i
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TUTORIAL: LINE INTEGRALS AND WORK
ii. What helped you determine the best choice of path? Discuss with your group members.
iii. What does this tell you about starting a problem involving a line integral?
Tutorial 4, Week 5
University of Colorado – Boulder
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